I'm working on analyzing the sensitivity of two matrix expressions. I'd like to formally show that one is more sensitive to perturbations in a covariance matrix C than the other.
We are given: $$\Delta_{\mathbf{A}}=(V_1-WK_1)K_1^TP(K_1K_1^TP+I)^{-1}$$ $$\Delta_{\mathrm{M}}=(V_1-WK_1)K_1^T(K_1K_1^T+C)^{-1}$$ Where:$V_1\in R^{d_{1}\times 1}$, $W\in R^{d_{1}\times d_{0}}$, $K_1\in R^{d_{0}\times 1}$, $C\in R^{d_{0}\times d_{0}}$, $d_0=6400$, $d_1=1600$, I is the identity matrix. The matrix P is constructed from the SVD of C as follows:$\{U,\Lambda,(U)^T\}=\mathrm{SVD}\left(C\right)$, define $\hat{U}$ as the matrix whose columns are the eigenvectors $u_i\in U\mid$ such that the corresponding singular value is "close to zero", $P=\hat{U}(\hat{U})^T$.
I want to rigorously show that $\Delta_{\mathbf{A}}$ is more sensitive to small changes in C than $\Delta_{\mathbf{M}}$.
My initial idea was to compute the derivatives $\frac{\partial\Delta_{\mathrm{A}}}{\partial C}$ and $\frac{\partial\Delta_{\mathrm{M}}}{\partial C}$ and compare their magnitudes (e.g., Frobenius norms). However, this fails because: The step: define $\hat{U}$ as the matrix whose columns are the eigenvectors $u_i\in U\mid$ such that the corresponding singular value is "close to zero" is a discontinuous set-valued map.
Is there a rigorous way to compare the sensitivity of $\Delta_{\mathbf{A}}$ and $\Delta_{\mathbf{M}}$ to perturbations in C. Any references or insights would be greatly appreciated!
